...The financial segment of this venture is a non-issue; I have and am willing to purchase entirely with my own savings, but under their name and official possession--correct me if this constitutes a straw purchase, as the laws governing this aspect of firearms are so convoluted that there are a million ambiguous cases...

It is legal for your parent to purchase a firearm from a dealer and give it to you as gift. But if they buy it for you with your money then that would constitute a straw purchase.